Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2022 / Paper 1 / 19 / a / iii / Solution

The summands repeat the six-term patternIn particular they do not tend to zero: the subsequence with $n=6k+1$ is always $\sqrt{3}/2$. The term test for divergence therefore shows that the series diverges. Although each block of six sums to zero, that only makes a subsequence of partial sums zero; it does not make the full sequence of partial sums convergent.